Question

How do you integrate ln(3x)?

Answer 1

Hint: We will first write the fact that $\int \ln (nx)dx=x\ln (nx)-x+C$. Now, we will just put in n = 3 and thus, we will get the required solution to the given question.

Complete Step by Step Solution:
We are given that we are required to integrate ln (3x).
We will first find the integration of ln (nx).
We can write this function as multiple of 1 and ln (nx).
Now, we need to find the value of $\int 1.\ln (nx)dx$.
Now using the ILATE rule, we have ln (nx) as the first function and 1 as the second function.
Now, we get:-
$\Rightarrow \int 1.\ln (nx)dx=x\ln (nx)-\int {\displaystyle \frac{n}{nx}}xdx+C$
We can write this as follows:-
$\Rightarrow \int \ln (nx)dx=x\ln (nx)-\int dx+C$
Now integrating dx on the right hand side in above equation, we will then obtain the following equation:-
$\Rightarrow \int \ln (nx)dx=x\ln (nx)-x+C$
Now, putting in n = 3, we will then obtain the following equation with us:-

$\Rightarrow \int \ln (3x)dx=x\ln (3x)-x+C$
Thus, we have the required answer.

Note:
The students must know the ILATE rule which has been mentioned above.
I stands for Inverse, L stands for Logarithmic, A stands for Algebraic, T stands for Trigonometric and E stands for exponential. This is the sequence in which we take the first function. Therefore, we have taken ln (3x) as the first function and 1 as the second function.
Also, note that if there are two function; first function being f (x) and the second function being g (x).
The integration of f (x) and g (x) is given by the following expression:-
$\Rightarrow \int f(x).g(x)dx=f(x)\int g(x)dx-\int \left\{\left({\displaystyle \frac{d}{dx}}f(x)\right)\int g(x)dx\right\}dx+C$
This is the formula that we used above.
The students must note that we may also use the formula directly.
We know that we have a formula for integration of ln (nx) which is given by the following expression:-
$\Rightarrow \int \ln (nx)dx=x\ln (nx)-x+C$
Now, putting in n = 3, we will then obtain the following equation with us:-
$\Rightarrow \int \ln (3x)dx=x\ln (3x)-x+C$
Thus, we have the required answer.